On the Hardness of Learning with Rounding over Small Modulus

We show the following reductions from the learning with errors problem (LWE) to the learning with rounding problem (LWR): (1) Learning the secret and (2) distinguishing samples from random strings is at least as hard for LWR as it is for LWE for efficient algorithms if the number of samples is no larger than O(q/Bp), where q is the LWR modulus, p is the rounding modulus, and the noise is sampled from any distribution supported over the set {−B, . . . , B}. Our second result generalizes a theorem of Alwen, Krenn, Pietrzak, and Wichs (CRYPTO 2013) and provides an alternate proof of it. Unlike Alwen et al., we do not impose any number theoretic restrictions on the modulus q. The first result also extends to variants of LWR and LWE over polynomial rings. As additional results we show that (3) distinguishing any number of LWR samples from random strings is of equivalent hardness to LWE whose noise distribution is uniform over the integers in the range [−q/2p, . . . , q/2p) provided q is a multiple of p and (4) the “noise flooding” technique for converting faulty LWE noise to a discrete Gaussian distribution can be applied whenever q = Ω(B √ m). All our reductions preserve sample complexity and have time complexity at most polynomial in q, the dimension, and the number of samples. ∗Part of this work done while authors were visiting IDC Herzliya, supported by the European Research Council under the European Union’s Seventh Framework Programme (FP 2007-2013), ERC Grant Agreement n. 307952. †[email protected]. Department of Computer Science and Engineering, Chinese University of Hong Kong. Work partially supported by RGC GRF grants CUHK410112 and CUHK410113. ‡[email protected]. Department of Computer Science and Engineering, Chinese University of Hong Kong. Work partially supported by RGC GRF grants CUHK410112 and CUHK410113. §[email protected]. Faculty of Mathematics, Ruhr-Universität Bochum. Supported by DFG Research Training Group GRK 1817/1. ¶[email protected]. Department of Computer Science, UCLA. ‖[email protected]. Efi Arazi School of Computer Science, IDC Herzliya, Israel. Work supported by ISF grant no. 1255/12 and by the European Research Council under the European Union’s Seventh Framework Programme (FP 2007-2013), ERC Grant Agreement n. 307952.